Balancer moving on a turntable

[Gabriel Maia]

Hi. The problem is the following:

We have a balancer initially at the end of a plank that spins around an axis at its center. The initial rotation period is T[itex]_{0}[/itex]. The balancer starts to walk with constant speed along the plank and we are asked to answer some questions about it.


a) What is the balancer's torque?

Since the force over the balancer is central it is consequently parallel to the radius and the torque is, therefore, zero. Right?


b) What are the angular momentum of the system (the balancer is to be trated as a point particle and the plank has momentum of inertia I=[itex]\frac{1}{12}[/itex]MD[itex]^{2}[/itex]) when the balancer reaches the center of the plank?

The angular momentum is given by [itex]\stackrel{\rightarrow}{L}[/itex]=[itex]\stackrel{\rightarrow}{r}[/itex]×[itex]\stackrel{\rightarrow}{p}[/itex]

where [itex]\stackrel{\rightarrow}{r}[/itex] is the vector connecting the position of the balancer to the axis of rotation. At the center of the plank r=0 so the total angular momentum is due exclusively to the plank. It is

L=I[itex]\omega[/itex]=[itex]\frac{1}{12}[/itex]MD[itex]^{2}[/itex][itex]\omega[/itex]

where [itex]\omega[/itex] is the angular velocity.

Right?

c) What is the rotation period in the iten b) situation?

We have that [itex]\omega[/itex]=2[itex]\pi[/itex]/T so

T=[itex]\frac{2\pi{I}}{L}[/itex]

I would like to express this independently of L but L is a function of T which cancels the other T. How can I determine it?

Thank you.

 

[haruspex]

a) What is the balancer's torque?

Since the force over the balancer is central it is consequently parallel to the radius and the torque is, therefore, zero. Right?

As the balancer moves in, what should be conserved? What does that imply about the rate of rotation? Assuming the plank has mass, what does that imply about a torque on the plank?

 

[Gabriel Maia]

The angular momentum should be conserved. As for the torque on the plank I would say that each element of the plank spins around the axis. So they are also subject to a central force and have torque zero.

 

[haruspex]

Yes, the angular momentum is conserved. So what will happen to the balancer+plank system as the balancer moves closer to the axis?

 

[Nickie]

I can confirm that your calculations and understanding are correct. In the first situation, the balancer's torque is indeed zero since the force is parallel to the radius. In the second situation, the angular momentum is solely due to the plank, as the balancer's position is at the center where the radius is zero. And in the third situation, you have correctly used the equation for angular velocity to calculate the rotation period. In order to express the period independently of L, you can use the equation for angular momentum in terms of angular velocity and moment of inertia, which is L = Iω. This allows you to solve for ω in terms of I and L, and then substitute that into the equation for period. I hope this helps clarify your calculations. Keep up the good work!